What are the divisors of 3657?

1, 3, 23, 53, 69, 159, 1219, 3657

There is a total of 8 positive divisors.
The sum of these divisors is 5184.
The arithmetic mean is 648.

8 odd divisors

1, 3, 23, 53, 69, 159, 1219, 3657

How to compute the divisors of 3657?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 3657 by each of the numbers from 1 to 3657 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

3657 / 1 = 3657 (the remainder is 0, so 1 is a divisor of 3657)
3657 / 2 = 1828.5 (the remainder is 1, so 2 is not a divisor of 3657)
3657 / 3 = 1219 (the remainder is 0, so 3 is a divisor of 3657)
...
3657 / 3656 = 1.0002735229759 (the remainder is 1, so 3656 is not a divisor of 3657)
3657 / 3657 = 1 (the remainder is 0, so 3657 is a divisor of 3657)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 3657 (i.e. 60.473134530963). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

3657 / 1 = 3657 (the remainder is 0, so 1 and 3657 are divisors of 3657)
3657 / 2 = 1828.5 (the remainder is 1, so 2 is not a divisor of 3657)
3657 / 3 = 1219 (the remainder is 0, so 3 and 1219 are divisors of 3657)
...
3657 / 59 = 61.983050847458 (the remainder is 58, so 59 is not a divisor of 3657)
3657 / 60 = 60.95 (the remainder is 57, so 60 is not a divisor of 3657)